Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.

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Unit Overview

Geometric thinking and spatial thinking are important in and of themselves, because they connect mathematics with the physical world. These two topics support the development of number and arithmetic concepts and skills. Geometry with its mathematical content, its roles in physical sciences, engineering, and many other subjects, and its strong aesthetic connections make essential learnings for all grades.

In grade 3, students extend the progression of geometry first introduced in kindergarten. In this unit, students analyze, compare, sort, and classify two-dimensional shapes through various problem-solving experiences. They learn to apply their ideas to entire classes of shapes rather than to individual shapes. They reason about, compose, and decompose polygons to make other polygons. Students’ understanding of the properties of shapes continues to be redefined, as they understand that shared attributes can define a larger category. They investigate quadrilaterals and recognize shapes that are not quadrilaterals. Students in Grade 3 also sort geometric figures and identify squares, rectangles, and rhombuses as quadrilaterals. Students also partition shapes into parts with equal areas and express the area of each part as a unit fraction of the whole. Classroom discussions are a vital part of this unit, and students should do the majority of the talking. The teacher’s role is to ask questions, to help students redefine their thinking, and to allow students to develop their understanding through whole-class, small group, and independent activities. Students should be encouraged to use proper mathematical vocabulary.

It is important to note that the Progressions for Geometry state: In this progression, the term “property” is reserved for those attributes that
indicate a relationship between components of shapes. Thus, “having parallel sides” or “having all sides of equal lengths” are properties. “Attributes” and “features” are used
interchangeably to indicate any characteristic of a shape, including properties, and other defining characteristics (e.g., straight sides) and non-defining characteristics (e.g.,
“right-side up”).

When implementing this unit, be sure to incorporate the Enduring Understandings and Essential Questions as the foundation for your instruction, as
appropriate.

Students should engage in well-chosen, purposeful, problem-based tasks. A good mathematics problem can be defined as any task or activity for which
the students have no prescribed or memorized rules or methods, nor is there a perception by students that there is a specific correct solution method (Hiebert et al., 1997). A
good mathematics problem will have multiple entry points and require students to make sense of the mathematics. It should also foster the development of efficient computations
strategies as well as require justifications or explanations for answers and methods.

Learning about Geometry does not progress in the same way as learning about number, where the size of the number gradually increases and new kinds
of numbers are considered later. Instead, students’ reasoning about Geometry develops through five sequential levels in relation to understanding spatial ideas. In order to
progress through the levels, instruction must be sequential and intentional. These levels were hypothesized by Pierre van Hiele and Dina van Hiele-Geldof.

Level 0: Visualization

Level 1: Analysis

Level 2: Informal Deduction

Level 3: Deduction

Level 4: Rigor

Attributes refer to any characteristic of a shape.

Through your discussions and interactions with students, emphasize reasoning about shapes and their attributes as emphasized in the Maryland Common
Core Standards, as opposed to simply identifying figures, which is typically only a vocabulary exercise. Definitions of geometric terms should connect to and evolve from classroom
experiences and discussions.

When partitioning shapes into parts with equal areas, express the area of each part as a unit fraction of the whole. Students are building on their understanding of fractional parts of the whole (the parts that result when the whole or unit has been partitioned into equal sized portions or fair
shares) from first and second grade.

Students should be encouraged to develop ideas and definitions about properties and classes of shapes based on their own concept of developments. Only after they have had ample time to build and discuss their own ideas should formal definitions be introduced.

In the U.S., the term “trapezoid” may have two different meanings. Research identifies these as inclusive and exclusive definitions. The inclusive definition states: A trapezoid is a quadrilateral with at least one pair of parallel sides. The exclusive definition states: A trapezoid is a quadrilateral with exactly one pair of parallel sides. With this definition, a parallelogram is not a trapezoid.(Progressions for the CCSSM: Geometry, The Common Core Standards Writing Team, June 2012.)

Enduring Understandings:

Geometry helps us understand the structure of space and the spatial relations around us.

Through geometry, we can analyze the characteristics and properties of two- and three-dimensional shapes as well as develop mathematical arguments concerning geometric relationships.

Geometry helps us develop and use rules for two- and three-dimensional shapes.

Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):

3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

3.NF.A.2 Developing an understanding of fractions as numbers is essential for future work with the number system. It is critical that students at this grade are able to place fractions on a number line diagram and understand them as a related
component of their ever-expanding number system.

3.MD.C.7 Area is a major concept within measurement, and area models must function as a support for multiplicative reasoning in grade 3 and beyond.

Possible Student Outcomes:

The student will be able to:

Understand concepts of area and relate area to multiplication and to addition.

Recognize perimeter as an attribute of plane figures and distinguish between linear and area measures.

Understand that shapes in different categories may share attributes and that the shared attributes can define a larger
category.

Recognize rhombuses, rectangles, and squares as examples of quadrilaterals, and draw examples of quadrilaterals that do not belong to any of these subcategories.

Partition shapes into parts with equal areas. Express the area of each part as a unit fraction of the whole.

Engage in well-chosen, purposeful, problem-based tasks that promote reasoning with shapes and their attributes.

Collaborate with peers in an environment that encourages student interaction and conversation that will lead to mathematical discourse about geometry.

Evidence of Student Learning:

The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.

Fluency Expectations and Examples of Culminating Standards:

Students fluently multiply and divide within 100.

Students fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the
relationship between addition and subtraction.

Common Misconceptions:

Students may:

Not realizing that when partitioning shapes in order to express the area of each part as a unit fraction, all parts must be equal.

Not realizing that some the properties of some shapes may place them into more than one larger category. For example, a square is a rectangle, parallelogram, and a quadrilateral.

The inability to create both regular and irregular polygons.

Not enough experience with the properties of shapes to recognize and correctly draw shapes having specified attributes, such as a given number of angles.

Interdisciplinary Connections:

Interdisciplinary connections fall into a number of related categories:

Literacy standards within the Maryland Common Core State Curriculum.

Science, Technology, Engineering, and Mathematics standards.

Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level
coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts,
among others.

Sample Assessment Items: The items included in this component will be aligned to the standards in the unit and will include:

This task presents students with some creative geometric ways to represent the fraction one half. The goal is both to appeal to students' visual intuition while also providing a hands on activity to decide whether or not two areas are equal. In order for students to be successful with this task, they need to understand that area is additive in the sense described in 3.MD.7d. Students should be given paper models of each picture which they can fold or cut and rearrange so as to help visualize why the shaded and unshaded areas are equal.

This task continues "3.G Which pictures represent half of a circle?" moving into more complex shapes where geometric arguments about cutting or work using simple equivalences of fractions is required to analyze the picture. In order for students to be successful with this task, they need to understand that area is additive in the sense described in 3.G.7.d. This task is meant for instructional purposes, and students should have access to multiple copies of the figures and be encouraged to experiment by cutting them and comparing the pieces. For the three pictures which do represent one half, this can be seen with a suitable rotation of the picture, which switches the shaded and unshaded portion, or by cutting out the shaded pieces and pairing them up with the unshaded ones so as to match in both shape and size. This task would not be appropriate for high-stakes assessment.

The purpose of this task is for students to use their understanding of area as the number of square units that covers a region (3.MD.6), to recognize different ways of representing fractions with area (3.G.2), and to understand why fractions are equivalent in special cases (3.NF.3.b). Determining the fraction of the area that is shaded for rectangles A-D in part (b) is increasingly complex. Rectangles E, F, and G show that there are many ways for of the area to be shaded blue, which implies that there are many ways to represent the fraction with area.

Rectangle H requires students to see the equivalence of two fractions, neither of which is a unit fraction. Students get a chance to demonstrate what they have learned in part (b) by generating their own representations of fractions in parts (c) and (d).

Interventions/Enrichments/PD: (Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)

Vocabulary/Terminology/Concepts: This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.

decomposing: breaking a number into two or more parts to make it easier with which to work.
Example: When combining a set of 5 and a set of 8, a student might decompose 8 into a set of 3 and a set of 5, making it easier to see that the two sets of 5 make 10 and then there are 3 more for a total of 13.
Decompose the number 4; 4 = 1+3; 4 = 3+1; 4 = 2+2
Decompose the number =

whole: In fractions, the whole refers to the entire region, set, or line segment which is divided into equal parts or segments.

numerator: the number above the line in a fraction; names the number of parts of the whole being referenced.
Example: I ate 3 pieces of a pie that had 5 pieces in all. So 3 out of 5 parts of a whole is written:
The 3 is the numerator, the part I ate. The 5 is the denominator, or the total number of pieces in the pie.

denominator: the number below the line in a fraction; states the total number of parts in the whole.
Example: I ate 3 pieces of a pie that had 5 pieces in all. So 3 out of 5 parts of a whole is written:
The 3 is the numerator, the part I ate. The 5 is the denominator, or the total number of pieces in the pie.

fraction of a region: is a number which names a part of a whole area.

Example:

Fraction of a set: is a number that names a part of a set.

Example:

Unit Fraction: a fraction with a numerator of one.

Examples:

benchmark fraction: fractions that are commonly used for estimation or for comparing other fractions.Example: Is ⅔ greater or less than ½?

improper fraction: a fraction in which the numerator is greater than or equal to the denominator.

mixed number: a number that has a whole number and a fraction.

area: the number of square units needed to cover a region. Examples:

tiling: highlighting the square units on each side of a rectangle to show its relationship to multiplication and that by multiplying the side lengths, the area can be determined. Example:

rectilinear figures: a polygon which has only 90° and possibly 270° angles and an even number of sides. Examples of Rectilinear Figures:

Part II – Instructional Connections outside the Focus Cluster

arrays: the arrangement of counters, blocks, or graph paper square in rows and columns to represent a multiplication or division equation. Examples:

equivalent fractions: different fractions that name the same part of a region, part of a set, or part of a line segment.

Blaisdell, Molly. If You Were a Quadrilateral.Notes: The author gets students thinking with statements such as, “If you were a quadrilateral, you would have four straight sides. You could be a checkerboard, a kite, or a yoga math. What else could you be if you were a quadrilateral”. The book offers opportunities for rich classroom discussions about quadrilaterals.

Emberly, Ed. Picture Pie.Notes: The author uses 4 simple shapes to show students how to draw various patterns, animals, and people.

Taylor- Cox, Jennifer. Sigmund Square Finds His Family.Notes: Even though he has order and friends in his life, things are not fair-square for Sigmund Square. With its many geometrical references and humorous storyline, the book can be used with students of all ages.

References:

------. 2000. Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.